Rings with simple Lie rings of Lie and Jordan derivations

2018 ◽  
Vol 17 (04) ◽  
pp. 1850078
Author(s):  
Ahmad Al Khalaf ◽  
Orest D. Artemovych ◽  
Iman Taha
Keyword(s):  
Noetherian Ring ◽  
Associative Ring ◽  
Lie Ring ◽  
Lie Rings ◽  
Lie Derivations ◽  
Torsion Free

Let [Formula: see text] be an associative ring. We characterize rings [Formula: see text] with simple Lie ring [Formula: see text] of all Lie derivations, reduced noncommutative Noetherian ring [Formula: see text] with the simple Lie ring [Formula: see text] of all derivations and obtain some properties of [Formula: see text]-torsion-free rings [Formula: see text] with the simple Lie ring [Formula: see text] of all Jordan derivations.

Derivations of differentially semiprime rings

2019 ◽  
Vol 12 (05) ◽  
pp. 1950079
Author(s):  
Ahmad Al Khalaf ◽  
Iman Taha ◽  
Orest D. Artemovych ◽  
Abdullah Aljouiiee
Keyword(s):  
Commutative Ring ◽  
Semiprime Ring ◽  
Prime Rings ◽  
Commutative Case ◽  
Lie Ring ◽  
Lie Rings ◽  
Semiprime Rings ◽  
Torsion Free

Earlier D. A. Jordan, C. R. Jordan and D. S. Passman have investigated the properties of Lie rings Der [Formula: see text] of derivations in a commutative differentially prime rings [Formula: see text]. We study Lie rings Der [Formula: see text] in the non-commutative case and prove that if [Formula: see text] is a [Formula: see text]-torsion-free [Formula: see text]-semiprime ring, then [Formula: see text] is a semiprime Lie ring or [Formula: see text] is a commutative ring.

FREE GROUPS OF OUTER COMMUTATOR VARIETIES OF GROUPS

2001 ◽  
Vol 64 (2) ◽  
pp. 423-435
Author(s):  
DANIEL P. GROVES
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Lie Ring ◽  
Lie Rings ◽  
Outer Commutator ◽  
Torsion Free ◽  
Group 1

If F is a free group, 1 < i [les ] j [les ] 2i and i [les ] k [les ] i + j + 1 then F/[γj(F), γi(F), γk(F)] is residually nilpotent and torsion-free. This result is extended to 1 < i [les ] j [les ] 2i and i [les ] k [les ] 2i + 2j. It is proved that the analogous Lie rings, L/[Lj, Li, Lk] where L is a free Lie ring, are torsion-free. Candidates are found for torsion in L/[Lj, Li, Lk] whenever k is the least of {i, j, k}, and the existence of torsion in L/[Lj, Li, Lk] is proved when i, j, k [les ] 5 and k is the least of {i, j, k}.

THE 2-GENERATOR RESTRICTED BURNSIDE GROUP OF EXPONENT 7

2002 ◽  
Vol 12 (04) ◽  
pp. 575-592 ◽  
Author(s):  
E. A. O'BRIEN ◽  
MICHAEL VAUGHAN-LEE
Keyword(s):  
Nilpotency Class ◽  
Lie Ring ◽  
Derived Length ◽  
Lie Rings ◽  
Associated Lie Ring

We report on our construction of a power-commutator presentation for R(2,7), the largest finite 2-generator group of exponent 7. Our calculations show that R(2,7) has order 720416, nilpotency class 28, and derived length 5. The calculations also imply that the associated Lie ring of R(2,7) satisfies relations which are not consequences of the multilinear identities which hold in the associated Lie rings of groups of exponent 7.

scholarly journals Dimension quotients of metabelian Lie rings

2017 ◽  
Vol 27 (02) ◽  
pp. 251-258
Author(s):  
Inder Bir S. Passi ◽  
Thomas Sicking
Keyword(s):  
Lower Central Series ◽  
Universal Enveloping Algebra ◽  
Central Series ◽  
Augmentation Ideal ◽  
Lie Ring ◽  
Ring Of Integers ◽  
Enveloping Algebra ◽  
Lie Rings

For a Lie ring [Formula: see text] over the ring of integers, we compare its lower central series [Formula: see text] and its dimension series [Formula: see text] defined by setting [Formula: see text], where [Formula: see text] is the augmentation ideal of the universal enveloping algebra of [Formula: see text]. While [Formula: see text] for all [Formula: see text], the two series can differ. In this paper, it is proved that if [Formula: see text] is a metabelian Lie ring, then [Formula: see text], and [Formula: see text], for all [Formula: see text].

Commutators in associative rings

1953 ◽  
Vol 49 (4) ◽  
pp. 590-594 ◽  
Author(s):  
M. P. Drazin ◽  
K. W. Gruenberg
Keyword(s):  
Associative Ring ◽  
Lie Ring ◽  
Addition And Subtraction ◽  
Associative Rings

Let R be an arbitrary associative ring, and X a set of generators of R. The elements of X generate a Lie ring, [X], say, with respect to the addition and subtraction in R, and the multiplication [a, b] = ab − ba. In this note we shall be concerned with the following question: if [X] is given to be nilpotent as a Lie ring, what does this imply about R?

scholarly journals Computing nilpotent quotients in finitely presented Lie rings

1997 ◽  
Vol Vol. 1 ◽  
Author(s):  
Csaba Schneider
Keyword(s):  
Abelian Group ◽  
Word Problem ◽  
Np Hard ◽  
Linear Combinations ◽  
Lie Ring ◽  
Lie Rings ◽  
Central Factor ◽  
Structure Theorems ◽  
International Audience ◽  
Finitely Presented

International audience A nilpotent quotient algorithm for finitely presented Lie rings over \textbfZ (and \textbfQ) is described. The paper studies the graded and non-graded cases separately. The algorithm computes the so-called nilpotent presentation for a finitely presented, nilpotent Lie ring. A nilpotent presentation consists of generators for the abelian group and the products expressed as linear combinations for pairs formed by generators. Using that presentation the word problem is decidable in L. Provided that the Lie ring L is graded, it is possible to determine the canonical presentation for a lower central factor of L. Complexity is studied and it is shown that optimising the presentation is NP-hard. Computational details are provided with examples, timing and some structure theorems obtained from computations. Implementation in C and GAP interface are available.

On Modules of Singular Submodule Zero

1971 ◽  
Vol 23 (2) ◽  
pp. 345-354 ◽  
Author(s):  
Vasily C. Cateforis ◽  
Francis L. Sandomierski
Keyword(s):  
Integral Domain ◽  
Associative Ring ◽  
Quotient Ring ◽  
Essential Extension ◽  
Singular Ideal ◽  
Free Modules ◽  
Torsion Free ◽  
Unitary Right

In this paper we generalize to modules of singular submodule zero over a ring of singular ideal zero some of the results, which are well known for torsion-free modules over a commutative integral domain, e.g. [2, Chapter VII, p. 127], or over a ring, which possesses a classical right quotient ring, e.g. [13, § 5].Let R be an associative ring with 1 and let M be a unitary right R-module, the latter fact denoted by MR. A submodule NR of MR is large in MR (MR is an essential extension of NR) if NR intersects non-trivially every non-zero submodule of MR; the notation NR ⊆′ MR is used for the statement “NR is large in MR” The singular submodule of MR, denoted Z(MR), is then defined to be the set {m ∈ M| r(m) ⊆’ RR}, where

scholarly journals Higher commutators, ideals and cardinality

1996 ◽  
Vol 54 (1) ◽  
pp. 41-54 ◽  
Author(s):  
Charles Lanski
Keyword(s):  
Finite Index ◽  
Associative Ring ◽  
Semiprime Ring ◽  
Torsion Free ◽  
The Ideal

For an associative ring R, we investigate the relation between the cardinality of the commutator [R, R], or of higher commutators such as [[R, R], [R, R]], the cardinality of the ideal it generates, and the index of the centre of R. For example, when R is a semiprime ring, any finite higher commutator generates a finite ideal, and if R is also 2-torsion free then there is a central ideal of R of finite index in R. With the same assumption on R, any infinite higher commutator T generates an ideal of cardinality at most 2cardT and there is a central ideal of R of index at most 2cardT in R.

scholarly journals Localization in non-noetherian group rings

1980 ◽  
Vol 21 (1) ◽  
pp. 151-163 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Abelian Group ◽  
Prime Ideal ◽  
Noetherian Ring ◽  
General Setting ◽  
Group Rings ◽  
Regular Local Ring ◽  
Characteristic P ◽  
Free Rank ◽  
Torsion Free Rank ◽  
Torsion Free

Let k be a field and G an Abelian group of finite torsion-free rank. Brewer, Costa and Lady [1, Theorem A] showed that if k has characteristic 0 then each Localization of the group algebra kG at a prime ideal is a regular local ring. They also showed (in the same theorem) that if k has characteristic p>0, then kG is locally Joetherian (i.e. each localization of kG at a prime ideal is a Noetherian ring) if and only if G is an extension of a finitely generated group by a torsion p′-group. The purpose of this note is to examine this theorem in a more general setting.

Derivations in differentially prime rings

2018 ◽  
Vol 17 (07) ◽  
pp. 1850129
Author(s):  
Ahmad Al Khalaf ◽  
Orest D. Artemovych ◽  
Iman Taha
Keyword(s):  
Commutative Ring ◽  
Prime Ring ◽  
Prime Rings ◽  
Commutative Case ◽  
Lie Ring ◽  
Lie Rings

Earlier properties of Lie rings [Formula: see text] of derivations in commutative differentially prime rings [Formula: see text] was investigated by many authors. We study Lie rings [Formula: see text] in the non-commutative case and shown that if [Formula: see text] is a [Formula: see text]-prime ring of characteristic [Formula: see text], then [Formula: see text] is a prime Lie ring or [Formula: see text] is a commutative ring.

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